In this paper we study multi-task kernel ridge regression and try to understand when the multi-task procedure performs better than the single-task one, in terms of averaged quadratic risk. In order to do so, we compare the risks of the estimators wit
h perfect calibration, the emph{oracle risk}. We are able to give explicit settings, favorable to the multi-task procedure, where the multi-task oracle performs better than the single-task one. In situations where the multi-task procedure is conjectured to perform badly, we also show the oracle does so. We then complete our study with simulated examples, where we can compare both oracle risks in more natural situations. A consequence of our result is that the multi-task ridge estimator has a lower risk than any single-task estimator, in favorable situations.
In this paper we study the kernel multiple ridge regression framework, which we refer to as multi-task regression, using penalization techniques. The theoretical analysis of this problem shows that the key element appearing for an optimal calibration
is the covariance matrix of the noise between the different tasks. We present a new algorithm to estimate this covariance matrix, based on the concept of minimal penalty, which was previously used in the single-task regression framework to estimate the variance of the noise. We show, in a non-asymptotic setting and under mild assumptions on the target function, that this estimator converges towards the covariance matrix. Then plugging this estimator into the corresponding ideal penalty leads to an oracle inequality. We illustrate the behavior of our algorithm on synthetic examples.
